<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>







<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="202_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="202_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:202_1_1"/>
( ( not <font color="Maroon">m</font> is <a href="abian.html#V1">even</a> &amp; not <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> ) or <font color="Maroon">m</font> is <a href="abian.html#V1">even</a> or <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> or  not <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or  not <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="202_1_1_1"><b>suppose </b></a><a NAME="E1:202_1_1_1"/><i><font color="Green">E39</font></i>: 
( not <font color="Maroon">m</font> is <a href="abian.html#V1">even</a> &amp; not <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">m'</font> = <font color="Maroon">m</font> as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="202_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:202_1_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K14" target="_self">.edges()</a> 
 ;<br/><b>then consider </b><font color="Maroon">x</font> being  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:202_1_1_1_1"/><i><font color="Green">E40</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">xaa1</font> = <font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> 1 as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="glib_001.html#E41"><i><font color="Green">Lemma143</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E5:202_1_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">xaa1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E6:202_1_1_1_1"/>
<font color="Maroon">xaa1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i><a class="txt" href="glib_001.html#E41"><i><font color="Green">Lemma143</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E7:202_1_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">xaa1</font></span>)</span> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">xaa1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">W</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E40"><i><font color="Green">Lemma142</font></i></a>, <a class="txt" href="glib_001.html#E41"><i><font color="Green">Lemma143</font></i></a>, , </i>;<br/><a NAME="E8:202_1_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">xaa1</font> &amp; <font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">xaa1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E9:202_1_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K14" target="_self">.edges()</a> 
 <b>by </b><i><a class="txt" href="glib_001.html#E42"><i><font color="Green">Lemma145</font></i></a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:202_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K14" target="_self">.edges()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font> <a href="glib_001.html#K14" target="_self">.edges()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="202_1_1_2"><b>suppose </b></a><a NAME="E1:202_1_1_2"/>
( <font color="Maroon">m</font> is <a href="abian.html#V1">even</a> or <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> or  not <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or  not <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:202_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K14" target="_self">.edges()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font> <a href="glib_001.html#K14" target="_self">.edges()</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:202"/>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K14" target="_self">.edges()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font> <a href="glib_001.html#K14" target="_self">.edges()</a> 
 ;<br/>


</div>
